On the group of homeomorphisms on R: A revisit

Kamaludheen Ali Akbar

India

Central University of Kerala

Department of Mathematics, School of Physical Sciences

T. Mubeena

India

University of Calicut

Department of Mathematics, School of Mathematics and Computational Science

Submitted: 2021-08-29

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Accepted: 2022-03-14

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Published: 2022-10-03

DOI: https://doi.org/10.4995/agt.2022.16143

Copyright (c) 2022 Applied General Topology

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Keywords:

Group of homeomorphisms, Normal subgroups, Dynamical systems, Fixed points, Conjugacy, bounded functions

Supporting agencies:

This research was not funded

Abstract:

 In this article, we prove that the group of all increasing homeomorphisms on R has exactly five normal subgroups, and the group of all homeomorphisms on R has exactly four normal subgroups. There are several results known about the group of homeomorphisms on R and about the group of increasing homeomorphisms on R ([2], [6], [7] and [8]), but beyond this there is virtually nothing in the literature concerning the topological structure in the aspects of topological dynamics. In this paper, we analyze this structure in some detail.

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References:

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M. Brin and G. Stuck, Introduction to Dynamical Systems, Cambridge University Press, 2002. https://doi.org/10.1017/CBO9780511755316

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A. G. O'Farrell, Conjugacy, involutions, and reversibility for real homeomorphisms, Irish Math. Soc. Bulletin 54 (2004), 41-52. https://doi.org/10.33232/BIMS.0054.41.52

S. Ulam and J. von Neumann, On the group of homeomorphisms of the surface of the sphere, (abstract), Bull. Amer. Math. Soc. 53 (1947), 506.

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